The point $(1,1,1)$ is rotated $180^\circ$ about the $y$-axis, then reflected through the $yz$-plane, reflected through the $xz$-plane, rotated $180^\circ$ about the $y$-axis, and reflected through the $xz$-plane.  Find the coordinates of the point now.
Solution: After $(1,1,1)$ is rotated $180^\circ$ about the $y$-axis, it goes to $(-1,1,-1).$

After $(-1,1,-1)$ is reflected through the $yz$-plane, it goes to $(1,1,-1).$

After $(1,1,-1)$ is reflected through the $xz$-plane, it goes to $(1,-1,-1).$

After $(1,-1,-1)$ is rotated $180^\circ$ about the $y$-axis, it goes to $(-1,-1,1).$

Finally, after $(-1,-1,1)$ is reflected through the $xz$-plane, it goes to $\boxed{(-1,1,1)}.$

[asy]
import three;

size(250);
currentprojection = perspective(6,3,2);

triple I = (1,0,0), J = (0,1,0), K = (0,0,1), O = (0,0,0);
triple P = (1,1,1), Q = (-1,1,-1), R = (1,1,-1), S = (1,-1,-1), T = (-1,-1,1), U = (-1,1,1);

draw(O--2*I, Arrow3(6));
draw((-2)*J--2*J, Arrow3(6));
draw(O--2*K, Arrow3(6));
draw(O--P);
draw(O--Q);
draw(O--R);
draw(O--S);
draw(O--T);
draw(O--U);
draw(P--Q--R--S--T--U,dashed);

label("$x$", 2.2*I);
label("$y$", 2.2*J);
label("$z$", 2.2*K);

dot("$(1,1,1)$", P, N);
dot("$(-1,1,-1)$", Q, SE);
dot("$(1,1,-1)$", R, dir(270));
dot("$(1,-1,-1)$", S, W);
dot("$(-1,-1,1)$", T, NW);
dot("$(-1,1,1)$", U, NE);
[/asy]